In this thesis we investigate results about convex combinations of unitaries in unital
J B* -algebras, the Jordan algebra analogues of C* -algebras. After giving background
material in chapter 1 we introduce the concept of unitary isotopes of J B* -algebras
in chapter 2 and develop their theory including identification of their centre. We
also show important unit aries for our results come from the polar decomposition
of invertible elements. In chapter 3 we investigate which elements are self-adjoint
in some isotope to start the development of the theory of convex combinations of
two or more unitaries. This leads us in chapter 4 to introduce and give examples
of a subclass of J B* -algebras in which the invertible elements are dense. We also
show that extreme points of the unit ball sufficiently close to the invertibles must be
unit aries and deduce that in the subclass, all extreme points are unitaries. In chapter
5 we look the relationships between the distance of an element to the invertibles or
unit aries and special types of convex combinations, for example those of unit aries
having all (but one) of the coefficients equal and those of unit ball elements only one
of which need be unitary. Finally in chapter 6 we investigate some possible further
developments and open problems